How to import a character into Unity with textures from BI?
Unity does not normally import textures along with meshes since this is in fact a poor design choice on your part, it's best to separate mesh from texture. Here is a document detailing what Unity can import from Blender with the .blend files.

Here is a post detailing the process of importing objects from Blender to Unity. 

Brief step by step:


Generate a UV map for your mesh
Make the texture for the object
Save your texture file to a separate file. .jpg should do, or .png for textures with a lot of similar colors/few colors overall.
Save it as either a .blend or .fbx file to import to Unity
Import your object to Unity
Browse to find your object
Find your material options for your mesh, and drag your texture file from the file viewer into the texture/albedo slot


REMEMBER: Unity has limited import options from Blender, so don't use any nodes or anything that's not basic. You'll have to do all that work in Unity since it's the engine that will render it all.

Hope this helps